Strictly speaking the below list is in order by sum of a+b rather than a+b+c, as evidenced by a+b+c=2016 preceding a+b+c=2010, to take the first such example. However, it does include all a+b under 10,000 and thus all a+b+c under 10,000. Only one a+b+c listed is above that value and thus there might exist unlisted entries above 10,000 and below 10,290, the only a+b+c listed that's over 10,000.

The two lowest values sought are 6 and 36.

a b c      a+b+c

2 2 2      6

9 18 9      36

32 32 16      80

28 84 28      140

70 105 35      210

162 162 54      378

65 260 65      390

144 288 72      504

273 364 91      728

126 630 126      882

266 665 133      1064

512 512 128      1152

469 603 134      1206

456 760 152      1368

455 1001 182      1638

217 1302 217      1736

793 854 183      1830

756 945 189      1890

448 1344 224      2016

737 1072 201      2010

730 1241 219      2190

729 1458 243      2430

417 1807 278      2502

728 1729 273      2730

1250 1250 250      2750

344 2408 344      3096

721 2060 309      3090

1120 1680 280      3080

702 2457 351      3510

211 3165 422      3798

665 2926 399      3990

1110 2590 370      4070

1705 2046 341      4092

604 3473 453      4530

1628 2849 407      4884

513 4104 513      5130

1136 3905 497      5538

2592 2592 432      5616

1040 4160 520      5720

386 4825 579      5790

2340 3276 468      6084

217 5642 651      6510

1617 4312 539      6468

2304 4608 576      7488

3354 3913 559      7826

730 6570 730      8030

1474 6633 737      8844

3185 5096 637      8918

3666 5170 658      9494

2268 6804 756      9828

4802 4802 686      10290

DefDbl A-Z

Dim crlf$

Private Sub Form_Load()

 Form1.Visible = True

 

 Text1.Text = ""

 crlf = Chr$(13) + Chr$(10)

 For tot = 2 To 10000

   For a = 1 To tot / 2

      DoEvents

       b = tot - a

       c4 = a * a * a + b * b * b

       c = Int(Sqr(Sqr(c4)) + 0.5)

       If c * c * c * c = c4 Then

         Text1.Text = Text1.Text & a & Str(b) & Str(c) & "      " & a + b + c & crlf

       End If

   Next a

 Next tot

 Text1.Text = Text1.Text & crlf & " done"

  

End Sub